Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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extension of automorphism of field to algebraically closed field

Suppose that F is a field contained in an algebraically closed field A. Prove that every automorphism of F can be extended to an automorphism of A.
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How would I show that $Z_7(t^9)$ subset $Z_7(t)$ isn't a normal extension where $Z_7(t)$ is a rational function field?

It's not clear to me what rational function fields are and the significance of normal extensions in the first place. $Z_7(t^9)$ seems smaller than $Z_7(t)$ so it makes sense one would be contained in ...
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Eisenstein generalization

Problem: Given a polynomial $f(x)=c_0+c_1\,x+\ldots+c_n\,x^n\in\mathbb{Z}[x]$, assume that exists $p\in\mathbb{Z}$ prime that satisfies: $p\,\nmid\,c_n$ $p\,\mid\,c_i,\;\forall i=0,\,\ldots,n-1$. ...
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468 views

Usage of finite fields or Galois fields in real world

I'm currently studying the theory of Galois fields. And I have a question, what practical usage of this finite fields? As stated in Wikipedia: Finite fields are important in number theory, ...
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two Non isomorphic root field-extension of the field.

Does there exist two non isomorphic minimal field extension ( root field) of $f= \frac {x^{64}-x}{x(x-1)} \in F_2[x]$ . I may be using wrong word here saying minimal field extension but in german ...
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$L/K$ field extension of degree $2^k$. Element $a$ in $L$ or $K$?

Let $L/K$ be a field extension of degree $2^k$, $k\in\mathbb{N}$. $a\in L$ and $f\in K[X]$ a polynomial of degree $d$ such that $f(a)=0$, $d$ odd. Show that $a \in K$. I know only the definition of a ...
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If $L/k$ and $\gcd(f(x),g(x))=h(x)$ in $k[x]$, then $\gcd(f(x),g(x))$ is also $h(x)$ in $L[x]$?

I am trying to prove this theorem: Let $L/k$ be field extension, and let $f(x),g(x)$ be two polynomials in $k[x]\setminus\{0\}$ and $h(x)=\gcd(f(x),g(x))$ in $k[x]$, then prove that the ...
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Inverse of the fundamental theorem of Galois Theory for finite extensions

Let $E/K$ be a finite Galois field extension. Then by the fundamental theorem of Galois theory, there is canonical bijection between the subgroups of $\mathrm{Gal}(E/K)$ and the intermediate field ...
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Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite?

Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite? Here $[\mathbb R:K]$ means the dimension of $\mathbb R$ as a $K$-vector space. What I have tried: If we can find ...
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What is the meaning of $K/F$ is a cyclic extension?

I have it that $K/F$ is a (finite) field extension, what is the definition of when $K/F$ is called cyclic ? I heard it while I studied Galois theory and it was defined as $K/F$ is called cyclic ...
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etale fundamental group of a product

Let $X,Y$ be noetherian connected schemes over an algebraically closed field $k$ and let $\overline{x},\overline{y}$ be geometric points on them. There is a canonical homomorphism $\pi_1(X \times ...
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Where is the calculation hiding in this proof about how $p$ splits in $\mathbb{Q}(\zeta_n)$?

I just worked through a proof in Daniel Marcus' book Number Fields that if $p\nmid n$, the inertial degree of any prime ideal of $\mathbb{Q}(\zeta_n)$ lying over $p$ is equal to the order of $p$ in ...
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An angle $\theta$ can be trisected if and only if $4x^3-3x+\cos\theta$ is reducible over $\mathbb{Q}(\cos\theta)$

I was given the above problem for homework. There is (what seems to be) a relevant proof in my textbook regarding the impossibility of trisecting $\pi/3$. In this proof, the identity $$\cos 3\theta = ...
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Real embedding of the splitting field of $X^3-2$

Does the splitting field of $X^3-2$ have a real embedding?
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49 views

Let $q=p^n$, $p$ prime, ¿Which $q$ satisfies $\mathbb{F}_{q^2}=\mathbb{F}_q(\sqrt{a})$?.

Let $q=p^n$, $p$ prime, ¿Which $q$ satisfies $\mathbb{F}_{q^2}=\mathbb{F}_q(\sqrt{a})$?. I don't know why is necessary a condition for $q$.
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Is the field extension $K(t):K$ normal? Is separable?

The field extension $K(t):K$ where $K$ is any field. I don't know how to apply definition of normal and separable in a transcendental case.
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searching the fixed field of an automorphism, and a primitive generator , in characteristic p.

Let $K$ be a field oh characteristic $p$. Let's take $\sigma \in \operatorname{Aut}(K(x),K)$ where $x$ is trascendental over $K$, where $\sigma(x)=x+1$. Find a primitive element of the fixed field of ...
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529 views

Computing the Galois group of $x^4+ax^2+b \in \mathbb{Q}[x] $

I want to compute the Galois group of some polynomials, but I want to see some examples first. For example this proposition could be helpful. I don't know how to prove it <.< Let's consider a ...
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1answer
558 views

Computing the Galois group of polynomials $x^n-a \in \mathbb{Q}[x]$

I have some problems with this exercise. I don't know if it can be done. Consider the polynomial $ x^n - a \in \mathbb{Q} $ Can I compute the Galois group of this over $\mathbb{Q}$? Maybe having a ...
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finite fields, a cubic extension on finite fields.

Let $q=p^n$ where $p$ is a prime number. For which $q$ is the extensión $\mathbb{F}_{q^3}/\mathbb{F}_q$ (i.e the extension of degree $3$) an extension of the form $\mathbb{F}_q(\root 3 \of \alpha )$ ...
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the proof of a proposition of galois theory

Let f(x) be a separable polynomial over the field K, with roots $r_1 , ... , r_n$ in it's splitting field F. Then f(x) is irreducible over K if and only if Gal(F/K) acts transitively on the roots of ...
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searching an explicit isomorphism of finite fields

Since all the finite field of $p^n$ elements are the splitting field of the separable polynomial $x^{p^n}-x$, all of them are isomphic. In particular if $f_1(x),f_2(x)$ are irreducible polynomials ...
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565 views

computing the galois group of a polynomial

Compute the Galois group of the splitting field of the polynomial $t^4-3t^2+4$ over $\mathbb{Q}$. I don't know how can I do this problem, the roots are very "ugly" maybe if I consider another basis ...
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splitting field extending an automorphism

Let $\varphi:K_1\to K_2$ be an isomorphism of fields. Let's consider a polynomial $p(x)\in K_1[x]$, and it's splitting field extension $ E_1/K_1 $, also the polynomial $\varphi(p(x))$ $\in K_2[x]$, ...
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How do we use Galois theory to show that an integral has no closed form? [duplicate]

Possible Duplicate: How can you prove that a function has no closed form integral? How do we use Galois theory to show that an integral has no closed form ? I know this is called ...
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1answer
146 views

What is the Galois group of $x^n + (x-1)^n $?

What is the Galois group of $x^n + (x-1)^n $ over the rationals in terms of the integer $n$ ? In case that is too hard , what is it for the first 20 integers ?
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What is the galois group of $x+3$ or $(x+1)(x+2)$ ? How about $A(x)B(x)$?

As the title says I wonder what the galois group of $x+3$ is. Or even if that exists ? Since $x+3 = 0$ has only one zero/element I assume its the trivial group ? And what is the galois group of ...
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Minimal polynomial of the form $\zeta_p+\frac{1}{\zeta_p}+\zeta_q+\frac{1}{\zeta_q }$?

We can calculate the minimal polynomial of $ 2cos(\frac{2\pi}{7})=\zeta_7+\frac{1}{\zeta_7}$ over Q as x^3+x^-2x-1 and simlary for $2cos(\frac{2\pi}{5})=\zeta_5+\frac{1}{\zeta_5 }$. Now my question ...
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Determining a Galois group without factoring

Let $f \in \mathbb{Q}[X]$ be irreducible and let $L$ be its splitting field. Can something be said about the Galois group of $L$ over $\mathbb{Q}$ without computing the roots of $f$ in $\mathbb{C}$?
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A proof of the fundamental theorem of symmetric polynomials

I'm reading Exploratory Galois Theory by John Swallow. On page 123 he gives the following remark / alternate proof of the fundamental theorem of symmetric polynomials: Let $K$ be a field and $L$ ...
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Characteristic 2

Show that for a field $L$ of characteristic 2 there exist quadratic equations wich cannot be solved by adjoining square roots of elements in the field $L$. Attempt: In $\mathbb{Z_2}$ adjoining all ...
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369 views

Compositum of abelian Galois extensions is also?

Suppose I have a field $k$ and two extensions $K/k$ and $L/k$ which are both abelian Galois extensions of $k$. Then (assuming $K$ and $L$ are both contained in some bigger field) is the compositum ...
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Serre's Topics in Galois Theory

I am supposed to write a running notes on Hilbert Irreducibility Theorem from Serre's Topics in Galois Theory as part of my master's Algebra course work. But I don't have any knowledge of Algebraic ...
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A basis of Galois extention $L/K(R)$

Ler $R$ be a normal domain with its quotient field $K$. Let $L$ be a finite Galois extention of $K$. Let $T$ be the integral closure of $R$ in $L$. Then we can take a basis $t_1,\cdots t_m$ of vector ...
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Application of Galois theory to cubic polynomial

Given a cubic polynomial $f(x) \in \mathbb{Z}[x_{1},\dots,x_{n}]$. Suppose that $f(x)$ does not factor into three linear polynomials but contains a linear factor. Is the linear factor defined over ...
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A prime in a separable extension splits completly iff it does so in the galois closure.

I just did the following exercise out of Neukirch's Algebraic Number Theory: "A prime ideal p of K is totally split in the separable extension L|K iff it is totally split in the Galois closure N|K of ...
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Galoisgroup $\operatorname{Gal}(K(\mu_n) / K) \subseteq (\mathbb{Z} / (n) )^*$

Let $K$ be a field and $\mu_n$ a primitive $n$-th root of unity. Then I can embed $\operatorname{Gal}(K(\mu_n) / K) \subseteq (\mathbb{Z} / (n) )^*$. For $K=\mathbb{Q}$ there would be even an ...
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Is every group a Galois group?

It is well-known that any finite group is the Galois group of a Galois extension. This follows from Cayley's theorem (as can be seen in this answer). This (linked) answer led me to the following ...
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Finite extensions of rational functions

I know that finite extensions of $\mathbb{C}(x)$ correspond to finite branched covers of $\mathbb{P}^1$, and this leads to an abstract characterization of the absolute Galois group of $\mathbb{C}(x)$ ...
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447 views

A polynomial whose Galois group is $D_8$

I need to construct such a polynomial, and more generally: given a group $G$, how can it be realized as a Galois group?
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How to prove $t^{23}+1$ irreducible in $F_p$?

I have tried to prove that $t^2+1$ is irreducible over $F_3$ by supposing to the contrary $t^2+1=(t+\alpha)(t+\beta)=t^2+(\alpha+\beta)t+\alpha\beta$. Then, $\alpha+\beta\equiv 0 \pmod 3, ...
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An explicit calculation of Galois group

This is a question requires to compute the Galois group of $X^4+1$ over $\mathbb{Q}$, $\mathbb{Q}(i)$, $\mathbb{F}_3$ and $\mathbb{F}_5$. Here is a brief of what I can think of. For the first two, ...
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269 views

Galois extension and Tensor product

The following theorem is proved in Bourbaki's algebra. They use the technique of Galois descent. I'd like to know the proof without using it if any. Theorem Let $K$ be a field. Let $\Omega/K$ be an ...
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In a quadratic extension, does the orthogonal complement of trace have a nice name?

The trace of an element in a quadratic extension is the sum of it with its conjugate. Is there a name for the difference between it and its conjugate?
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Primitive Element for Field Extension of Rational Functions over Symmetric Rational Functions

A rational function $f$ in $n$ variables is a ratio of $2$ polynomials, $$f(x_1,...x_n) = \frac{p(x_1,...x_n)}{q(x_1,...x_n)}$$ where $q$ is not identically $0$. The function is called symmetric if ...
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What is this quadratic form as a invariant of Galois Extensions?

Suppose that $E/F$ is a Galois extension and viewing E as a vector space over $F$, then quadratic from $Tr_F^{E}(\alpha^2)(\alpha\in E)$ carries some information of the extension. My question is that, ...
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How to show $\mathbb{Q}(\alpha^{4})=\mathbb{Q}(\alpha)$?

From Berkeley Problems in Mathematics, Spring 1999, Problem 17. Let $f(x)\in \mathbb{Q}[x]$ be an irreducible polynomial of degree $n\ge 3$. Let $L$ be the splitting field of $f$, and let $\alpha\in ...
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Irreducible polynomials with integer coefficients over Q

Suppose p(x) is an irreducible polynomial over Q of degree n, with integer coefficients. If p(x) has two roots r1 and r2 satisfying r1r2 = 5, prove that n is even. Attempt at solution: Because the ...
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98 views

Technical question on integral ring extensions

Let $A$ be an integral domain, integrally closed in its field of quotients $K$ and let $L$ be a finite Galois extension of $K$ with group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $p$ be ...
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Integral Galois Extensions - Proposition 2.4 (Lang)

My question refers to the proof of Proposition 2.4, p. 341 in Lang's Algebra. Here is the context: Let $A$ be an integral domain, integrally closed in its field of quotients $K$ and let $L$ be a ...